An Unhappy End
Two engines, at opposite ends of a single track railway line, set off towards one another just as a fly, sitting on the front of one of the engines, sets off flying along the railway line...
Problem
Two engines are 1000 metres apart at the ends of a single track railway line.
They set off towards one another at 10 metres/second.
Just at that moment, a fly sitting on the front of one of the engines sets off and starts to fly along the railway line at 25 metres/second.
The fly eventually meets the front buffer of the other engine which by now has travelled some way along the track.
The fly immediately turns round and flies at its usual speed towards the first engine.
When it gets there it turns round again and flies back towards the other engine, and so on, and so on...
How far will the fly have travelled before the unhappy end?
Getting Started
It may be useful to know how much travelling time the fly has available.
Student Solutions
We had many correct solutions to this one including a spreadsheet from Brian, Michael, Jon, James, Eli and Sam from Queen's College Junior School in Taunton, showing how far the trains and fly had travelled at each second after the trains set off.
Thomas from Darley Dale Primary School explained his solution like this:
First, you had to work out the distance travelled before the two trains collided, thus causing the 'Unhappy Ending'. This worked out as 500m (half of 1000m.).
The time it took the trains to travel this distance at 10m per second was 50 seconds.
Therefore, the fly had 50 seconds before the trains collided.
As the fly flew at 25m per second, it travelled 1250m (50 times 25) before it met its 'unhappy ending.'
Joe from Haslingfield worked it out as follows:
The trains are 1000m apart.
They travel at the same speed which means they both cover 500m before they crash.
The fly travels at 2.5 times the speed of the trains so it covers 2.5 times the distance the trains do. Therefore the calculation you need to work out is 500m x 2.5 which equals 1250m.
Well done to you all and to everyone else who answered this problem correctly: Steven from Whybridge Junior School, Robert from Ardingly College Junior School, Charlotte from Winterton Juniors, Helena, Sarah and Tom from Colyton Grammar School, Samuel from Geneva English School, Charlie and Ian from William Lovell School, Natalie, Matilda and Laura, Elle and Cat from The Mount School, Mandy Appleyard from Ashlyns School, Lawrence from Beecroft Public School (NSW, Ausralia), Salma from Balwearie High School, Andrew and Yuming from The Perse School for Boys, Alex from St Edwards School, Matthew from Waverley Christian College, Sean from St. Andrew's School...
Teachers' Resources
Why do this problem?
This problem only requires some simple understanding of the relationship between time, distance and speed, but it will require clear thinking and insight to solve. It provides a good experience of the mathematical 'lightbulb' moment when a seemingly tricky problem is solved using a clever trick or alternative viewpoint.
Possible approach
Key questions
- How long will it be until the fly first hits a train?
- How long will it be until the trains collide?
Possible extension
Possible support
As a warm up, you might try the simpler problem Bike Ride